Answer:
[tex]\left(x+1\right)\left(x+4\right)\left(x+7\right)[/tex]
Step-by-step explanation:
First, let's simplify the given expression:
[tex](x + 4)^3 - 9x - 36 = 0[/tex] (Simplify the exponent)
= [tex]x^3 + 12x^2 + 48x + 64 - 9x - 36 = 0[/tex] (Combine like terms)
= [tex]x^3 + 12x^2 + 39x + 28 = 0[/tex] (According to the Rational Root Theorem, we can try to factor out 1 since it is a possible factor)
= [tex]\left(x+1\right)\frac{x^3+12x^2+39x+28}{x+1}[/tex] (Hey, it worked! Now we can simplify this)
= [tex](x + 1)x^2+11x+28[/tex] (We can now factor x² + 11x + 28 as a regular quadratic trinomial)
= [tex]\left(x+1\right)\left(x+4\right)\left(x+7\right)[/tex]
